Problem 23 - Precalculus with Calculus Previews

Question

In Problems \(23-28,\) use the discriminant to identify the conic without actually graphing. $$ x^{2}-3 x y+y^{2}=5 $$

Step-by-Step Solution

Answer

The conic is a hyperbola.

Step 1: Identify the general form of the conic

The given equation is already in the general form of a conic section: \[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]For the equation \( x^2 - 3xy + y^2 = 5 \), the coefficients are \( A = 1 \), \( B = -3 \), and \( C = 1 \).

Step 2: Calculate the discriminant

The discriminant of a conic section given by the equation \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \) is calculated as:\[ D = B^2 - 4AC \]Substitute \( A = 1 \), \( B = -3 \), and \( C = 1 \) into the discriminant formula:\[ D = (-3)^2 - 4(1)(1) = 9 - 4 = 5 \]

Step 3: Identify the type of conic based on the discriminant

To identify the type of conic section:- If \( D > 0 \), the conic is a hyperbola.- If \( D = 0 \), the conic is a parabola.- If \( D < 0 \), the conic is an ellipse.Here, \( D = 5 > 0 \), indicating the conic is a hyperbola.

Key Concepts

Discriminant

The discriminant is a crucial mathematical tool that helps us determine the nature of conic sections without graphing them. For equations in the form \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \), the discriminant \( D \) is calculated using the formula:

• \( D = B^2 - 4AC \)

By substituting the coefficients from the equation into this formula, we can solve for \( D \). The value of the discriminant tells us the type of conic section:

• If \( D > 0 \), the conic section is a hyperbola.
• If \( D = 0 \), the conic section is a parabola.
• If \( D < 0 \), the conic section is an ellipse.

This method provides a straightforward way to classify conic sections just by analysing their coefficients.

Hyperbola

A hyperbola is a type of conic section represented by an equation where the discriminant \( D > 0 \). This means hyperbolas happen when the curve is more stretched compared to ellipses or circles.Hyperbolas can be visualized as twin curves that mirror each other across a central point, and they tend to curve outwards. They appear in many different contexts, from architecture to physics.The simplest form of a hyperbola is given by \( rac{x^2}{a^2} - rac{y^2}{b^2} = 1 \) or its rotated version. Whenever you encounter a conic section with \( D > 0 \), it is safe to classify the shape as a hyperbola.

General Form of a Conic

The general form of a conic section provides a standardized way of expressing conic equations, which is:

• \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \)

This form is comprehensive, as different combinations of the coefficients \( A, B, C, D, E, \) and \( F \) can describe any conic section: circles, ellipses, parabolas, and hyperbolas.Understanding this generalized form is essential because it allows us to manipulate and analyse conics in a systematic manner. By examining the relationships and values of these coefficients, particularly the discriminant, we can classify and understand the geometry of the shape described by the equation.

Conic Section Classification

Conic section classification is a method to differentiate between the types of curves formed in a plane by intersecting a cone with a plane. The classification is mainly based on the value of the discriminant in the general conic equation.Different types of conic sections include:

• Circle: A special case of an ellipse when \( A = C \) and \( B = 0 \).
• Ellipse: Formed when \( D < 0 \) with \( A eq C \).
• Parabola: Occurs when \( D = 0 \).
• Hyperbola: The curve when \( D > 0 \), indicating two open, diverging paths.

This classification guides us not only in understanding the specific shape and properties of the curves but also in predicting their behavior without extensive graphing.